I have viewed lots of pics taken with both lenses but have not ever shot the same subject with both lenses side-by-side in a direct comparison.
I have owned a 80-400 for almost a decade now.
Out of all the images I have viewed, I have not seen a compelling reason, in terms of image quality, to buy the 70-300 in preference to the 80-400. They are very similar in performance with the 80-400 possibly having a small edge when not at full stretch
The 80-400 is very sharp up to 300mm. Between 300 and 400 it loses a little bit of sharpness but can still produce very good results. the extra reach of the 80-400 makes a big difference for avian photography and small game animals. Since you expressed interest in avian photography, I would pretty much disregard the 70-300 unless you can reliably get to within 10 feet (for small birds) or 30 feet for larger ones
The 80-400 is much heavier and longer. It is easily used for handheld shots and is one of the most portable 400mm lenses available, but still not a lens you would want to walk around with all day. It has a foot for tripod use. Unfortunately the foot is not very comfortable for carrying the combination.
The 80-400 is a D-AF lens, having a mechanical AF mechanism. this means it can only be used as a manual focus lens on the entry level bodies such as D40, D60, D3x00, D5x00. anachronistically, the D50 does have the screwdriver AF mechanism. the lens is also very slow to focus and it hunts a lot so while it is great for static subjects, I have not had any success with birds in flight using this lens.
Alternatives in similar price bracket are the Sigma 50-500 (aka Bigma) and the newer Sigma 150-500. Both will focus faster and published tests say these lenses are similarly sharp but in images I have seen, the Nikon 80-400 has superior colour saturation and contrast, if not sharpness. If you find a second-hand Sigma 170-500 then ignore it. I have never seen a decently sharp image from this old lens.
To get anything significantly better, you have to spend 2 or three times more - so I am not selling mine any time soon. Have had a number of great shots with it over the years.